is normalized provided that |up |2 + |vp |2 = 1 for all p.

If, instead of momentum states, position states had been used we would analo-

gously have encountered creation operators which create fermions at de¬nite posi-

tions, for example

a† 1 a† 2 . . . a† N |0 = |x1 § x2 § · · · § xN . (1.92)

x x x

Creation operators of di¬erent representations are related through their transforma-

tion functions. For position and momentum we have, according to Eq. (1.5),

dp

e’ p·x |p

i

|x dp |p p|x

= = (1.93)

(2π )3/2

and therefore the relationship

dp

a† = e’ p·x a† .

i

(1.94)

x p

3/2

(2π )

In accordance with tradition, instead of using the notation a† we shall introduce

x

ψ † (x) = a† . (1.95)

x

To obtain the inverse relation we use

dx

e x·p |x

i

|p dx |x x|p

= = (1.96)

(2π )3/2

1.3. Fermi ¬eld 21

and we have the relationships between the operators creating particles with de¬nite

position and momentum

dp dx

ψ † (x) = e’ p·x a† a† = e p·x ψ † (x) .

i i

, (1.97)

p p

(2π )3/2 (2π )3/2

Taking the adjoint we obtain analogously for the annihilation operators or quantum

¬elds

dp dx

e p·x ap e’ p·x ψ(x) .

i i

ψ(x) = , ap = (1.98)

(2π )3/2 (2π )3/2

How one prefers to keep track of factors of 2π in the above Fourier transfor-

mations, is, of course, a matter of taste. With the above convention determined by

the fundamental choice of Eq. (1.5), no such factors appear in the fundamental anti-

commutation relation, Eq. (1.78). If the fermions are con¬ned to a box of volume V

we shall use (guided by our preference for ¬elds to have the same dimensions as wave

functions)

1 1

p·x

dx e’ p·x

i i

ψ(x) = √ ap = √

e ap , ψ(x) , (1.99)

V V V

p

leaving the fundamental anti-commutation relation Eq. (1.78)

{ap , a† } = δp,p , (1.100)

p

where the discrete allowed momentum values are speci¬ed by the boundary condition

for the states, say periodic boundary conditions.

One readily veri¬es, as a consequence of the analogous Eq. (1.78), or by using

Eq. (1.97) and Eq. (1.98), the fundamental anti-commutation relations for Fermi

¬elds in the position representation17

{ψ(x), ψ † (x )} = δ(x ’ x ) (1.101)

and

{ψ(x), ψ(x )} = 0 = {ψ † (x), ψ † (x )} . (1.102)

For equal position the latter two equations have, as a consequence, ψ(x) ψ(x) = 0

and ψ † (x) ψ † (x) = 0, expressing the exclusion principle: no two identical fermions

can occupy the same position.

For the N -particle basis state with particles at the indicated locations we shall

also use the notation

† † †

¦x1 x2 ...xN = ψx1 ψx2 · · · ψxN |0 (1.103)

17 If the particles represented by the ¬elds have internal degrees of freedom, say spin, we have

{ψ(x, σz ) , ψ† (x , σz )} = δσ z ,σ z δ(x ’ x ) .

Often the notation ψσ z (x) = ψ(x, σz ) is used.

22 1. Quantum ¬elds

where |0 denotes the vacuum state for the fermions.

As stressed, any complete set of single-particle states, not just the position or

momentum states, could have been employed. For example in the absence of transla-

tion invariance and using the single-particle energy eigenstates we have analogously

for the quantum ¬elds

—

ψ(x) = x|» a» = ψ» (x) a» , a» = dx ψ» (x) ψ(x) , (1.104)

» »

where ψ» (x) are the orthonormal eigenstates of a single-particle Hamiltonian.

Instead of characterizing the quantum statistics of a collection of fermions in

terms of the antisymmetry of their state vectors, which as we have seen is a bit

messy or at least requires a substantial amount of indices-writing, it is now taken

care of by the simple anti-commutation relations for the creation and annihilation

operators. The price paid for this enormous simpli¬cation is of course that the

operators now are operators on a super-space, the multi-particle space. As shown in

the next section, the implementation for bosons is identical to the above except that

the quantum statistics is taken care of by the commutation relations of the creation

and annihilation operators.

Exercise 1.8. For N non-interacting spin one-half fermions, an ideal Fermi gas, the

ground state is obtained from the vacuum state according to

⎛ ⎞

|G0 = ⎝ a† ⎠ |0 , (1.105)

p,σ

σ,|p|<pF

i.e all the states below the Fermi energy, F = p2 /2m, are occupied in accordance

F

with Pauli™s exclusion principle, and all states above are empty for the case of the

ground state. Pictorially, the ground state is that of a ¬lled sphere in momentum

space, the Fermi sea, with the Fermi surface separating occupied and unoccupied

states.

Show that the one-particle Green™s function or density matrix becomes

†

Gσ (x ’ x ) ≡ G0 |ψσ (x)ψσ (x )|G0

3n sin kF |x ’ x | ’ kF |x ’ x | cos kF |x ’ x |

= , (1.106)

(kF |x ’ x |)3

2

where n is the density of the fermions, and kF = pF / , and in the considered three

dimensions kF = 3π 2 n. The considered amplitude speci¬es the overlap between the

3

state where a particle with spin σ at position x has been removed from the ground

state and the state where a particle with spin σ at position x has been removed

from the ground state. Or equivalently, it speci¬es the amplitude for transition to

the ground state of the state where a particle with spin σ at position x has been

removed from the ground state and subsequently a particle with spin σ has been

added at position x.

1.4. Bose ¬eld 23

At small spatial separation

(kF |x ’ x |)2

n

† †

G0 |ψσ (x)ψσ (x 1’

)|G0 (1.107)

2 10

and at x = x it counts the density of fermions per spin at the position in question.

Show that the pair correlation function is related to the one-particle density ma-

trix according to

§

⎪ n2

⎨2 σ =σ

†

†

G0 |ψσ (x)ψσ (x )ψσ (x )ψσ (x)|G0 = (1.108)

⎪ n2

© ’ Gσ (x ’ x ) σ = σ.

2

2

Interpret the result and note in particular the anti-bunching of non-interacting fermions:

the avoidance of identical fermions to be at the same position in space, a repulsion

solely due to the exchange symmetry, the exclusion principle at work in real space.

So far the creation operators are just a kinematic gadget giving an equivalent way

of describing the N -particle state space for arbitrary N , since for example

a† 1 a† 2 · · · a† N |0 = |p1 § p2 § · · · § pN (1.109)

p p p

speci¬es the basis states in terms of the creation operators and the vacuum state.

In Chapter 2, we shall show how operators representing physical quantities can be

expressed in terms of the creation and annihilation operators, and thereby realize in

Chapter 3 their usefulness in describing quantum dynamics in the most general case

where the number of particles is not conserved. But ¬rst we consider the kinematics

for the case where the identical particles are bosons.

1.4 Bose ¬eld

The bose particle creation operator, a† , is introduced according to its action on the

p

basis states of Eq. (1.45)

a† |p1 ∨ p2 ∨ · · · ∨ pN ≡ |p ∨ p1 ∨ p2 ∨ · · · ∨ pN (1.110)

p

and the adjoint operates according to

N

ap |p1 § · · · § pN p|pn |p1 § · · · ( no pn ) · · · § pN , (1.111)

=

n=1

i.e. it annihilates a particle in state p, and is referred to as the bose annihilation

operator. As previously noted, the derivation is equivalent to the antisymmetric case.

Since no minus signs ever occur, the bose creation and annihilation operators sat-

isfy the commutation relations (the analogous equations to Eq. (1.76) and Eq. (1.77)

are now subtracted to give the following result)

[ap , a† ] = p |p (1.112)

p

24 1. Quantum ¬elds

and

[a† , a† ] = 0 = [ap , ap ] . (1.113)

p

p

We note that, according to the equation for bosons analogous to Eq. (1.77), the

operator a† ap counts the number of particles in state p

p

a† ap |p1 ∨ · · · ∨ pN = np |p1 ∨ · · · ∨ pN (1.114)

p

where np denotes the number of particles in momentum state p in the basis state

|p1 ∨ · · · ∨ pN , i.e. the number of pi s which are equal to p, and the operator np =

a† ap is referred to as the number operator for state or mode p. In contrast to the

p

case of fermions, the boson number operators have besides the eigenvalue 0 all natural

numbers as eigenvalues. As in the case of fermions, the total set of momentum state

number operators, {np }p , thus constitute a complete set of commuting operators

giving rise to a representation as discussed in Section 1.5.

Quite analogous to the case of fermions, creation and annihilation operators with

respect to position can be introduced. For the N -particle basis state with particles

at the indicated locations we have

† † †

¦x1 x2 ... xN = ψx1 ψx2 . . . ψxN |0 , (1.115)

where |0 denotes the vacuum state for the bosons.

Kinematically, independent boson ¬elds are assumed to commute, and bose ¬elds

commute with fermi ¬elds (at equal times).

Though already stated, the expression for the resolution of the identity is not

of much practical use; the job has been taken over by the creation and annihilation

operators, we include it for completeness. The resolution of the identity in the multi-

particle space takes the form (and identically for fermions by using the antisymmetric

states)

∞

1

|p1 ∨ p2 ∨ · · · ∨ pN p1 ∨ p2 ∨ · · · ∨ pN |

1 =

N!

N =0 p1 ,p2 ,...,pN

∞

1

|p1 ∨ p2 ∨ · · · ∨ pN p1 ∨ p2 ∨ · · · ∨ pN | ,

=

n1 ! · · · nN !

p1 ¤p2 ¤···

N =0

(1.116)

where the term N = 0 denotes the projection operator onto the vacuum, |0 0|.

Exercise 1.9. Compact notation encompassing both bosons and fermions can some-

times be convenient. Writing an anti-commutator {A, B} ≡ [A, B]+ , the double val-

ued variable, s = ±, comprises both anti-commutators and commutators, [A, B]s ,

and distinguishes the two types of quantum statistics. Show that

[np , a† ]s = a† , [np , ap ]s = s ap . (1.117)

p p

1.4. Bose ¬eld 25

1.4.1 Phonons

The bose ¬eld does not occur only in connection with the elementary bosonic par-

ticles of the standard model, but can be useful in describing collective phenomena

such as the long wave length oscillations of the ions in say a metal or a semicon-

ductor, and we turn to see how this comes about. The Hamiltonian describing the

ions of mass M and density ni in a crystal lattice is given by the kinetic energy

term for the ions and an e¬ective ion“ion interaction determined by the screened

Coulomb interaction. Expanding the e¬ective ion“ion interaction potential to low-

est, quadratic, order, neglecting anharmonic e¬ects and thus only accounting for

small oscillations of the ions, the Hamiltonian can be diagonalized by an orthogo-

nal transformation rendering it equivalent to that of a set of independent harmonic

oscillators. In this long wave length description, the background dynamics can be

described by a continuum limit quantum ¬eld, the quantum displacement ¬eld, u(x),

a coarse-grained description of the ionic displacements at position x. For longer than

interatomic distance, the screened Coulomb interaction is e¬ectively a delta function,

Ve¬ (x ’ x ) = Z 2 /2N0 δ(x ’ x ), and together with the kinetic energy of the back-

ground ions, the background Hamiltonian functional valid for small displacements

then becomes18

M n i c2

1 2

(∇x · u(x))2

Hb = dx (Π(x)) + , (1.118)

2M ni 2

where the components of the momentum density and the displacement ¬eld inherit

the canonical commutation relations of the ions

δ±β δ(x ’ x )

[Π± (x) , uβ (x)] = (1.119)

i

and the sound velocity is given by

Zn Zm 2

c2 = = v (1.120)

3M F

2N0 M

where n = Zni is the equilibrium electron density and m the electron mass. We

note that the longitudinal sound velocity is typically smaller by a factor of 100 than

the Fermi velocity, vF . The continuum description of the oscillations of the in fact

discretely located ions appeared because the ions were assumed to exhibit only small

oscillations.

The Hamiltonian describing the dynamics of the background is in fact just a set

of harmonic oscillators, as obtained by diagonalizing the Hamiltonian. Introducing

the normal mode operators

1/2

k · ck

2M ni ωk

where ck + c† = dx e’ix·k u(x)

ak = , (1.121)

’k

V k

V

18 For details of these arguments, starting from the quantum mechanics of the individual ions and

then taking the continuum limit, we refer the reader to, for example, chapter 10 of reference [1].

26 1. Quantum ¬elds

or

1

(ck eik·x + c† e’ik·x )

u(x) = (1.122)

’k

V

k=0,|k|<kD

the background Hamiltonian becomes the free longitudinal phonon Hamiltonian

1

a† ak +

Hph = Hb = ωk (1.123)

k

2

|k|¤kD

with linear dispersion ωk = c |k|, and the operators satisfy the harmonic oscillator

normal mode commutation relations

[ak , a† ] = δk,k [a† , a† ] = 0 ,

, [ak , ak ] = 0 (1.124)

k kk

inherited from the canonical commutation relations for the position and momentum

operators of the individual ions. A quantum of an oscillator, a quantized sound mode,

is referred to as a phonon. In the Debye model, the lattice vibrations are assumed

to have linear dispersion all the way to the cut-o¬ wave vector kD .

However, instead of the above quantum mechanical argument, we can also here

take the opportunity to discuss the classical ¬eld theory of oscillations in an isotropic

elastic medium, and then obtain the corresponding quantum ¬eld theory by quan-

tizing the dynamics of the normal modes. This trick can then be elevated to give us

the quantum theory of the electromagnetic ¬eld.

1.4.2 Quantizing a classical ¬eld theory

As an example of quantizing a classical ¬eld theory we consider an elastic isotropic

medium of volume V speci¬ed by its longitudinal sound velocity c and mass density

ρ. In terms of the displacement ¬eld, u(x, t), describing the displacement of the

background matter at position x at time t, we have for small displacements

δnb (x, t)

= ’ ∇ · u(x, t) , (1.125)

ni

where δnb (x, t) is the deviation of the medium density from the average density

ni . Newton™s equation and the continuity equation leads for small δnb (x, t) to this

density disturbance satisfying the wave equation

1 ‚2

x’ 2 δnb (x, t) = 0 (1.126)

c ‚t2

or the dynamics of the elastic medium is equivalently, through the principle of least

action, described by the Lagrange functional

2

ρ ‚u(x, t)

’ c2 (∇ · u(x, t))2

™

L[u, u] = dx . (1.127)

2 ‚t

In accordance with the assumed isotropy of the elastic medium, it exhibits no

shear or vorticity, sustaining only longitudinal waves

∇x — u(x, t) = 0 , k — uk (t) = 0 , uk (t) . (1.128)

k

1.4. Bose ¬eld 27